Let us now place a source in the path of a uniform flow. The
stream function and the velocity potential for the resulting flow
are given by adding the two stream functions and velocity
potentials as follows,

(4.89)

(4.90)

One of the interesting features to determine for the resulting
force is the stagnation point of the flow, i.e., where the
velocity goes to zero.

One could calculate this from the equations. It is clear that for
this flow the stagnation point will occur on the x-axis. The
location can be arrived at purely intuitionally. The source
produces a radial flow of magnitude

while the uniform flow produces a velocity of U in the positive
x-direction. When these two cancel out at a point we have the
stagnation point. A negative radial flow that can cancel the
uniform flow is possible only to the left of the x-axis, say at
x = -b. Hence,

leading to

(4.91)

At x = -b, we have
and r = b. Substituting these values
in the expression for , i.e., Eqn. 4.89 we get the
value of at the stagnation point to be

(4.92)

An equation to the streamline passing through the stagnation
point, i.e., stagnation streamline is obtained as follows,

But hence

leading to

(4.93)

Figure 4.24: Flow about Rankine Half Body

The streamlines for this flow are sketched in Fig.4.24.
It is clear that we can make the stagnation streamline the solid
body. In fact any streamline of a flow can be treated as a solid
body since there is no flow across it. In the present example if
we ignore the streamlines inside the "body" we have described the
flow about a solid body given by Eqn. 4.25. This body
is referred to as a Rankine Half Body as it is "open" at
the right hand end.

Limits of for this body are 0 and . At these
values we have y approaching , which is called the Half
Width of the body.

which enables us to calculate the pressure. Usually in aerodynamic
applications involving significant velocities and pressures any
contribution due to elevation changes is negligible. The equation
for pressure assumes a simple form,